Understanding Equations of Circles

This video tutorial covers the equations of circles, focusing on converting between standard and general forms. It explains how to write circle equations given the center and radius, and how to graph these equations. The tutorial also delves into advanced techniques for converting equations and foiling to achieve general form. Key concepts include understanding the standard form of a circle equation, converting between forms, and graphing circles.

What is the standard form of a circle's equation?

• x^2 + y^2 + ax + by + c = 0

• ax^2 + by^2 = c

• (x - h)^2 + (y - k)^2 = r^2

• x^2 + y^2 = r^2

If a circle has a center at (3, -2) and a radius of 4, what is its equation?

• (x + 3)^2 + (y + 2)^2 = 16

• (x - 3)^2 + (y + 2)^2 = 16

• (x + 3)^2 + (y - 2)^2 = 16

• (x - 3)^2 + (y - 2)^2 = 16

How do you determine the center of a circle from its equation in standard form?

• By calculating the average of the x and y terms

• By looking at the coefficients of x and y

• By taking the opposite signs of the constants in the equation

• By finding the midpoint of the diameter

What is the radius of a circle if its equation is x^2 + y^2 = 144?

• 144

• 12

• 72

• 24

When converting a circle's equation from general to standard form, what is the first step?

• Factor the equation

• Complete the square

• Group x and y terms

• Move the constant to the left side

What is the result of completing the square for the expression x^2 - 8x?

• (x + 8)^2

• (x + 4)^2

• (x - 4)^2

• (x - 8)^2

In the equation 2x^2 + 2y^2 = 16x + 4y + 20, what should be done first to simplify it?

• Add 20 to both sides

• Divide all terms by 2

• Subtract 16x from both sides

• Multiply all terms by 2

What is the general form of a circle's equation?

• x^2 + y^2 = r^2

• x^2 + y^2 + ax + by = r^2

• (x - h)^2 + (y - k)^2 = r^2

• ax^2 + by^2 + cx + dy + e = 0

How do you convert a circle's equation from standard to general form?

• By completing the square

• By expanding the squared terms

• By factoring the equation

• By simplifying the coefficients

What is the center of a circle with the equation (x - 4)^2 + (y + 1)^2 = 7?

• (4, 1)

• (-4, 1)

• (4, -1)

• (-4, -1)